Question

[Kuramajiva]

Câu 1: Cho 2 số dương a,b,c. Chứng minh rằng:\( \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}<\sqrt\frac{a}{b+c}+\sqrt\frac{b}{c+a}+\sqrt\frac{c}{a+b}\)

Answer 1

[Nguyễn Việt Lâm]

\(VT=\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}\)

\(VT< \dfrac{a+c}{a+b+c}+\dfrac{b+a}{a+b+c}+\dfrac{c+b}{a+b+c}=2\)

\(VP=\dfrac{a}{\sqrt{a\left(b+c\right)}}+\dfrac{b}{\sqrt{b\left(c+a\right)}}+\dfrac{c}{\sqrt{c\left(a+b\right)}}\)

\(VP\ge\dfrac{2a}{a+b+c}+\dfrac{2b}{a+b+c}+\dfrac{2c}{a+b+c}=2\)

\(\Rightarrow VP>VT\) (đpcm)